Lennard-Jones Clusters Research Articles

An Efficient Growth Pattern Algorithm (GrowPAL) for Cluster Structure Prediction.

Identifying the lowest energy isomers in large clusters is a major challenge. Here, we introduce the Growth Pattern Algorithm (GrowPAL), a new approach that generates initial seeds composed of n+1 atoms from the system with n atoms through an interstitial-type addition (I-type) mechanism. We evaluated the effectiveness of GrowPAL on Lennard-Jones (LJ) clusters with up to n = 80 atoms, verifying the algorithm's ability to find challenging minima such as LJ38 and the partially icosahedral LJ69 with fewer optimizations than existing methods. In addition, we discuss the advantages and limitations of GrowPAL using our deconstruction scheme, which identifies "forebears" structures to study growth pathways. Having evaluated the strengths and weaknesses of GrowPAL, we employed it to explore Sutton-Chen clusters containing 5 to 80 atoms, uncovering three new lowest energy forms. We then applied GrowPAL to boron clusters containing 8 to 15 atoms, successfully identifying all reported minima. Overall, GrowPAL offers a practical solution for efficiently identifying global minima in hierarchical systems, thereby reducing computational costs.

Clustering molecular energy landscapes by adaptive network embedding

In order to efficiently explore the chemical space of all possible small molecules, a common approach is to compress the dimension of the system to facilitate downstream machine learning tasks. Towards this end, we present a data-driven approach for clustering potential energy landscapes of molecular structures by applying recently developed Network Embedding techniques to obtain latent variables defined through the embedding function. To scale up the method, we also incorporate an entropy sensitive adaptive scheme for hierarchical sampling of the energy landscape, based on Metadynamics and Transition Path Theory. Taking into account the kinetic information implied by the energy landscape of a system, we can interpret dynamical node-node relationships in reduced dimensions. We demonstrate the framework through Lennard-Jones clusters and a human DNA sequence.

Connecting the unstable region of the entropy to the pattern of the Fisher zeros map

Phase transitions are one of the most interesting natural phenomena. For finite systems, one of the concerns in the topic is how to classify a specific transition as a being of first, second, or even of a higher order according to the Ehrenfest classification. The partition function provides all the thermodynamic information about the physical systems, and a phase transition can be identified using the complex temperature where it is equal to zero. In addition, the pattern of zeros in the complex temperature plane can provide evidence of the transition order. This manuscript presents an analytical and simulational study connecting the microcanonical analysis of the unstable region of the entropy to the canonical partition function zeros. We show that, for the first-order transition, the zeros accumulate uniformly in a vertical line on the complex inverse temperature plane as discussed in previous works. We illustrate our calculations using a 147 particles Lennard-Jones cluster.

Monte Carlo-Simulated Annealing and Machine Learning-Based Funneled Approach for Finding the Global Minimum Structure of Molecular Clusters.

Understanding the physical underpinnings and geometry of molecular clusters is of great importance in many fields, ranging from studying the beginning of the universe to the formation of atmospheric particles. To this end, several approaches have been suggested, yet identifying the most stable cluster geometry (i.e., global potential energy minimum) remains a challenge, especially for highly symmetric clusters. Here, we suggest a new funneled Monte Carlo-based simulated annealing (SA) approach, which includes two key steps: generation of symmetrical clusters and classification of the clusters according to their geometry using machine learning (MCSA-ML). We demonstrate the merits of the MCSA-ML method in comparison to other approaches on several Lennard-Jones (LJ) clusters and four molecular clusters-Ser8(Cl-)2, H+(H2O)6, Ag+(CO2)8, and Bet4Cl-. For the latter of these clusters, the correct structure is unknown, and hence, we compare the experimental and simulated fragmentation patterns, and the fragmentation of the proposed global minimum matches experiments closely. Additionally, based on the fragmentation of the predicted betaine cluster, we were able to identify hitherto unknown neutral fragmentation channels. In comparison to results obtained with other methods, we demonstrated a superior ability of MCSA-ML to predict clusters with high symmetry and similar abilities to predict clusters with asymmetrical structures.

Structure search method for atomic clusters based on the dividing rectangles algorithm.

The Dividing Rectangles (DIRECT) algorithm is a deterministic optimization method to explore optimal solutions by repeatedly dividing a given hyperrectangle search space into subhyperrectangles. Herein, we propose a structure search method for atomic clusters based on the DIRECT algorithm in combination with a gradient-based local optimizer to enable an efficient structure search in high-dimensional search spaces. We use the Z-matrix representation for defining the hyperrectangle search space, in which the bond lengths, bond angles, and dihedral angles specify a cluster structure. To evaluate its performance, we applied the proposed method to the Lennard-Jones clusters and two kinds of real atomic clusters with many metastable structures, i.e., phosphorus and sulfur clusters, and compared the results with those of conventional methods. The proposed method exhibits a higher efficiency than random search and a comparable efficiency to basin hopping.

Crystal Polymorphism Induced by Surface Tension.

We use classical density functional theory (cDFT) to calculate fluid-solid surface tensions for fcc and bcc crystals formed by hard spheres and Lennard-Jones (LJ) particles. For hard spheres, our results show that the recently introduced "explicitly stable" functionals perform as well as the state of the art, and for both interaction potentials, our results compare well to simulation. We use the resulting bulk and interfacial energies for LJ to parametrize a capillary model for the free energy of small solid clusters and thereby determine the relative stability of bcc and fcc LJ clusters. We show a crossover from bcc to fcc stability as cluster size increases, thus providing insight into long-standing tension between simulation results and theoretical expectations. We also confirm that the bcc phase in contact with a vapor is unstable, thus extending earlier zero-temperature results. Our Letter demonstrates the potential of cDFT as an important tool in understanding crystallization and polymorphism.

A robust and memory-efficient transition state search method for complex energy landscapes.

Locating transition states is crucial for investigating transition mechanisms in wide-ranging phenomena, from atomistic to macroscale systems. Existing methods, however, can struggle in problems with a large number of degrees of freedom, on-the-fly adaptive remeshing and coarse-graining, and energy landscapes that are locally flat or discontinuous. To resolve these challenges, we introduce a new double-ended method, the Binary-Image Transition State Search (BITSS). It uses just two states that converge to the transition state, resulting in a fast, flexible, and memory-efficient method. We also show that it is more robust compared to existing bracketing methods that use only two states. We demonstrate its versatility by applying BITSS to three very different classes of problems: Lennard-Jones clusters, shell buckling, and multiphase phase-field models.

Neural Network for Principle of Least Action.

The principle of least action is the cornerstone of classical mechanics, theory of relativity, quantum mechanics, and thermodynamics. Here, we describe how a neural network (NN) learns to find the trajectory for a Lennard-Jones (LJ) system that maintains balance in minimizing the Onsager–Machlup (OM) action and maintaining the energy conservation. The phase-space trajectory thus calculated is in excellent agreement with the corresponding results from the “ground-truth” molecular dynamics (MD) simulation. Furthermore, we show that the NN can easily find structural transformation pathways for LJ clusters, for example, the basin-hopping transformation of an LJ38 from an incomplete Mackay icosahedron to a truncated face-centered cubic octahedron. Unlike MD, the NN computes atomic trajectories over the entire temporal domain in one fell swoop, and the NN time step is a factor of 20 larger than the MD time step. The NN approach to OM action is quite general and can be adapted to model morphometrics in a variety of applications.

Ab initio metadynamics determination of temperature-dependent free-energy landscape in ultrasmall silver clusters.

Ab initio metadynamics enables the extraction of free-energy landscapes having the accuracy of first-principles electronic structure methods. We introduce an interface between the PLUMED code that computes free-energy landscapes and enhanced-sampling algorithms and the Atomic Simulation Environment (ASE) module, which includes several ab initio electronic structure codes. The interface is validated with a Lennard-Jones cluster free-energy landscape calculation by averaging multiple short metadynamics trajectories. We use this interface and analysis to estimate the free-energy landscape of Ag5 and Ag6 clusters at 10, 100, and 300K with the radius of gyration and coordination number as collective variables, finding at most tens of meV in error. Relative free-energy differences between the planar and non-planar isomers of both clusters decrease with temperature in agreement with previously proposed stabilization of non-planar isomers. Interestingly, we find that Ag6 is the smallest silver cluster where entropic effects at room temperature boost the non-planar isomer probability to a competing state. The new ASE-PLUMED interface enables simulating nanosystem electronic properties under more realistic temperature-dependent conditions.

Mapping metastability of Lennard-Jones clusters by maximum vibrational frequency

We study the structure-stability relationship of the Lennard-Jones (LJ) clusters from a point of view of vibrations. By assuming the size up to $N=1610$, we demonstrate that the $N$-dependence of the maximum vibrational frequency reflects the geometry of the core (the interior of cluster) that will determine the overall geometry of the cluster. This allows us to identify the formation of non-icosahedral structures for $N\le 150$, the vacancy formation at the core for $N\ge 752$, and the transition from icosahedral to decahedral structures at $N = 1034$. We apply the maximum frequency analysis to classify metastable clusters for $19\le N \le 39$, where transformation pathways between different structures are visualized, and the energy barrier height is estimated simultaneously.

Visualizing Energy Landscapes through Manifold Learning

Energy landscapes provide a conceptual framework for structure prediction, and a detailed understanding of their topological features is necessary to develop efficient methods for their exploration. The ability to visualise these surfaces is essential, but the high dimensionality of the corresponding configuration spaces makes this difficult. Here we present Stochastic Hyperspace Embedding and Projection (SHEAP), a method for energy landscape visualisation inspired by state-of-the-art algorithms for dimensionality reduction through manifold learning, such as t-SNE and UMAP. The performance of SHEAP is demonstrated through its application to the energy landscapes of Lennard-Jones clusters, solid-state carbon, and the quaternary system C+H+N+O. It produces meaningful and interpretable low-dimensional representations of these landscapes, reproducing well known topological features such as funnels, and providing fresh insight into their layouts. In particular, an intrinsic low dimensionality in the distribution of local minima across configuration space is revealed.

Development of a Structural Comparison Method to Promote Exploration of the Potential Energy Surface in the Global Optimization of Nanoclusters.

A structural comparison method (SCM) was created to quantify the structural diversity of nanoclusters and was implemented into a global optimization algorithm to evaluate structural diversity between generated clusters on the fly and promote exploration of the potential energy surface. The SCM evaluated topological differences between clusters using the common neighbor analysis and provided a numerical measure of similarity between the two clusters. The SCM was implemented into a genetic algorithm by integrating it into a new structure + energy fitness operator such that structural diversity of clusters in the population and their energies were used to assign fitness values to clusters. The efficiency of the genetic algorithm with this new fitness operator was benchmarked against several Lennard-Jones clusters (LJ38, LJ75, and LJ98) known to be difficult cases for global optimization algorithms. For LJ38 and LJ75, this new structure + energy fitness operator performed equally well or better than the energy fitness operator. However, the efficiency of locating the global minimum of LJ98 decreased using this new structure + energy fitness operator. Further analysis of the genetic algorithm with this fitness operator showed that the algorithm did indeed promote exploration of the PES of LJ98 as desired but hindered refinement of clusters, preventing it from locating the global minimum even if the energy funnel of the global minimum had been located.

Simulated annealing with adaptive cooling rates.

As one of the most robust global optimization methods, simulated annealing has received considerable attention with many variations that attempt to improve the cooling schedule. This paper introduces a variant of molecular dynamics-based simulated annealing that is useful for optimizing atomistic structures, and makes use of the heat capacity of the system, determined on the fly during optimization, to adaptively control the cooling rate. This adaptive cooling approach is demonstrated to be more computationally efficient than classical simulated annealing when applied to Lennard-Jones clusters. The increase in efficiency is approximately a factor of two for clusters with 25-40 atoms, and improves as the size of the system increases.

Optimization of Lennard-Jones clusters by particle swarm optimization with quasi-physical strategy

Abstract The goal of Lennard-Jones (LJ) clusters optimization is to find the minimum value of the potential function of a cluster and thereby determine the stable configuration of the cluster. It is essentially a completely inseparable multimodal global optimization problem, and using the traditional particle swarm algorithm to solve it often results in local convergence, which means that the solution accuracy of the algorithm is not high. Thus, in this study, we develop a LJ algorithm using a particle swarm optimization (PSO) method and a physical approach to improve the solution accuracy. In this quasi-physical strategy (QPS), the particle swarm algorithm is used to simulate the real atomic structure and incorporates the interatomic force to construct a convergence model so that the algorithm performs well in both global and local space. The potential energy functions of a variety of LJ cluster systems are selected as test functions, and the improved PSO algorithm (QPS-PSO) is analyzed and compared with a competitive swarm optimizer, cooperative coevolution PSO, and differential-group cooperative coevolution, variable-length PSO for feature selection, heterogeneous comprehensive learning PSO, ensemble PSO and cooperative coevolution with differential optimization. The results show that the PSO algorithm for LJ clusters using the proposed QPS has noticeably superior solution accuracy, especially in high-dimensional spaces.

Funnel hopping Monte Carlo: An efficient method to overcome broken ergodicity.

Monte Carlo simulations are a powerful tool to investigate the thermodynamic properties of atomic systems. In practice, however, sampling of the complete configuration space is often hindered by high energy barriers between different regions of configuration space, which can make ergodic sampling completely infeasible within accessible simulation times. Although several extensions to the conventional Monte Carlo scheme have been developed, which enable the treatment of such systems, these extensions often entail substantial computational cost or rely on the harmonic approximation. In this work, we propose an exact method called Funnel Hopping Monte Carlo (FHMC) that is inspired by the ideas of smart darting but is more efficient. Gaussian mixtures are used to approximate the Boltzmann distribution around local energy minima, which are then used to propose high quality Monte Carlo moves that enable the Monte Carlo simulation to directly jump between different funnels. We demonstrate the method's performance on the example of the 38 as well as the 75 atom Lennard-Jones clusters, which are well known for their double funnel energy landscapes that prevent ergodic sampling with conventional Monte Carlo simulations. By integrating FHMC into the parallel tempering scheme, we were able to reduce the number of steps required significantly until convergence of the simulation.

Defining, Calculating, and Converging Observables of a Kinetic Transition Network.

Kinetic transition networks (KTNs) of local minima and transition states are able to capture the dynamics of numerous systems in chemistry, biology, and materials science. However, extracting observables is numerically challenging for large networks and generally will be sensitive to additional computational discovery. To have any measure of convergence for observables, these sensitivities must be regularly calculated. We present a matrix formulation of the discrete path sampling framework for KTNs, deriving expressions for branching probabilities, transition rates, and waiting times. Using the concept of the quasi-stationary distribution, a clear hierarchy of expressions for network observables is established, from exact results to steady-state approximations. We use these results in combination with the graph transformation method to derive the sensitivity, with respect to perturbations of the known KTN, giving explicit terms for the pairwise sensitivity and discussing the pathwise sensitivity. These results provide guidelines for converging the network, with respect to additional sampling, focusing on the estimates obtained for the overall rate coefficients between product and reactant states. We demonstrate this procedure for transitions in the double-funnel landscape of the 38-atom Lennard-Jones cluster.

A flexible and adaptive grid algorithm for global optimization utilizing basin hopping Monte Carlo.

Global optimization is an active area of research in atomistic simulations, and many algorithms have been proposed to date. A prominent example is basin hopping Monte Carlo, which performs a modified Metropolis Monte Carlo search to explore the potential energy surface of the system of interest. These simulations can be very demanding due to the high-dimensional configurational search space. The effective search space can be reduced by utilizing grids for the atomic positions, but at the cost of possibly biasing the results if fixed grids are employed. In this paper, we present a flexible grid algorithm for global optimization that allows us to exploit the efficiency of grids without biasing the simulation outcome. The method is general and applicable to very heterogeneous systems, such as interfaces between two materials of different crystal structures or large clusters supported at surfaces. As a benchmark case, we demonstrate its performance for the well-known global optimization problem of Lennard-Jones clusters containing up to 100 particles. Despite the simplicity of this model potential, Lennard-Jones clusters represent a challenging test case since the global minima for some "magic" numbers of particles exhibit geometries that are very different from those of clusters with only a slightly different size.

Stochastic approximation Monte Carlo with a dynamic update factor.

We present a Monte Carlo algorithm based on the stochastic approximation Monte Carlo (SAMC) algorithm for directly calculating the density of states. The proposed method is stochastic approximation with a dynamic update factor (SAD), which dynamically adjusts the update factor γ_{t} during the course of the simulation. We test this method on a square-well fluid and a 31-atom Lennard-Jones cluster and compare the convergence behavior of several related Monte Carlo methods. We find that both the SAD and 1/t-Wang-Landau (1/t-WL) methods rapidly converge to the correct density of states without the need for the user to specify an arbitrary tunable parameter t_{0} as in the case of SAMC. SAD requires as input the temperature range of interest, in contrast with 1/t-WL, which requires that the user identify the interesting range of energies. The convergence of the 1/t-WL method is very sensitive to the energy range chosen for the low-temperature heat capacity of the Lennard-Jones cluster. Thus, SAD is more powerful in the common case in which the range of energies is not known in advance.

Unbiased fuzzy global optimization of Lennard-Jones clusters for N ≤ 1000.

We propose a fuzzy global optimization (FGO) algorithm to identify the lowest-energy structure of nanoclusters. In contrast to traditional methods implemented in the real space, FGO utilizes mostly the discrete space in a fuzzy search framework. Starting from random initial configurations, we carry out directed Monte Carlo and surface Monte Carlo in the discrete space to obtain low-energy candidate clusters and make real-space local optimizations finally to get the real global minimum structure. The performance of FGO is demonstrated in a large set of standard Lennard-Jones (LJ) clusters with up to 1000 atoms. All the putative global minima reported in the literature are successfully obtained with a low scaling of CPU time with cluster size, and new global minimum structures for LJ clusters with 894, 974, and 991 atoms are identified. Due to the unbiased nature, FGO can potentially deal with the global optimization of other nanomaterials with high efficiency and reliability.

Description of unimolecular reaction rates of Lennard-Jones clusters

The unimolecular rate constant of Lennard-Jones clusters with N = 54 − 56 atoms has been investigated in constant energy molecular dynamics simulations and compared with the detailed balance predictions. Decays were found to be statistical and absolute rate constants agree quantitatively with detailed balance predictions. The level densities required for the comparison of simulated and calculated rate constants were obtained from separate simulations using the integrated microcanonical temperature. The quantitative agreement between simulations and theory requires that the classical degeneracy of the ground state is included into the level densities.